🇨🇳 Chinese Remainder Theorem Calculator
Solve systems of modular congruences
How to Use This Calculator
Enter First Congruence
Input the remainder a₁ and modulus m₁ for the first congruence x ≡ a₁ (mod m₁).
Enter Second Congruence
Input the remainder a₂ and modulus m₂ for the second congruence x ≡ a₂ (mod m₂).
View Solution
See the solution modulo the product of m₁ and m₂, with verification.
Chinese Remainder Theorem
Solve: x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂)
Where m₁ and m₂ are coprime (GCD = 1)
Example: x ≡ 2 (mod 3), x ≡ 3 (mod 5)
Solution modulo 15 (3×5): x = 8
Verification: 8 mod 3 = 2 ✓, 8 mod 5 = 3 ✓
Complete solution: 8 + 15k for any integer k
About Chinese Remainder Theorem Calculator
The Chinese Remainder Theorem Calculator solves systems of modular congruences. Given two congruences with coprime moduli, it finds a unique solution modulo their product.
When to Use This Calculator
- Number Theory: Solve modular arithmetic problems
- Cryptography: Understand RSA and modular arithmetic
- Computer Science: Learn number system conversions
- Competitions: Solve math competition problems
Why Use Our Calculator?
- ✅ Instant Solution: Find results immediately
- ✅ Verification: Check both congruences are satisfied
- ✅ Educational: Learn CRT concepts
- ✅ Completely Free: No registration required
Requirements
- Moduli m₁ and m₂ must be coprime (GCD = 1)
- Solution exists and is unique modulo m₁×m₂
- All solutions are congruent modulo the product
Frequently Asked Questions
What is the Chinese Remainder Theorem?
A method to solve systems of modular congruences when the moduli are coprime. It finds a unique solution modulo the product of the moduli.
Why must moduli be coprime?
For a unique solution to exist, the moduli must have GCD = 1. Otherwise, there may be no solution or multiple solutions.
What if the system has no solution?
This happens when the moduli are not coprime or when the remainders are incompatible. The calculator will alert you.